In this resubmission of a 5-year competing renewal grant application, we have carefully addressed the concerns raised in the previous review. The roadmap of this project from the beginning has been and continues to be the development, optimization, validation, and evaluation of 2D, 3D to 4D corrective image reconstruction methods for SPECT and PET in order to provide and demonstrate significant improvement in image quality for clinical diagnosis. Use and development of a variety of supporting methodologies, experiments and clinical trials have been and are essential in achieving this goal. Past accomplishments include pioneering and innovative work notably the development of the widely popular 4D computer generated NCAT phantom, accurate and efficient simulation of ECT data, 2D, 3D, and 4D corrective image reconstruction methods that have been licensed by a major commercial vendor, and task-based optimization and evaluation methods using populations of subjects that include anatomical and physiological variations. Our main hypothesis in the proposed project is that substantial further improvements in ECT image quality will be obtained from the use of 3D and 4D corrective image reconstruction methods, including corrections for non- clinical factors as well as patient involuntary motion, specifically respiratory motion. We propose to continue our ground-breaking work through further development of the 4D NCAT phantom for more realistic modeling of normal and abnormal respiratory anatomy and function. We will also extend our existing corrective image reconstruction methods from 2D and 3D to 4D with the goal of improving the detection of abnormalities. An important innovation is the proposed development and use of task-based evaluation methods that use mathematical observers and populations of phantoms that include anatomical and physiological variations that realistically model those found in clinical data. Several important clinical ECT applications including myocardial perfusion SPECT, oncological SPECT using 123I labeled agents and oncological lung and liver PET using 18F labeled FDG have been chosen as examples to evaluate the clinical efficacy of the 4D corrective image reconstruction methods. Realistic simulated imaging data from populations of 4D NCAT phantoms with anatomical and physiological variations, experimental data from physical phantoms, and clinical data from patient studies will be used. Results of the evaluation studies using simulated data from populations of phantoms and mathematical and human observers will be compared to those obtained using clinical data and trained physicians. In addition to contributing to the improvement of clinical diagnosis through the development of 4D corrective image reconstruction methods for ECT, the proposed research will make significant contributions to the understanding of observer performance in clinical evaluation of medical imaging systems and techniques, provide more realistic simulation tools, and demonstrate their the utility as possible replacement for costly clinical trials.The goal of the project is to develop methods to give doctors better images of the heart and cancers using SPECT and PET scanners. The research makes use of state-of-the-art scientific methods to find the best possible ways to give much clearer images than those provided by existing methods. To make sure the new methods are really an improvement, we will check them carefully using computer generated images, then images from experiments, and finally to images from approved patient studies.